Strain engineering of electronic properties and anomalous valley hall conductivity of transition metal dichalcogenide nanoribbons

Strain engineering is a powerful technique for tuning electronic properties and valley degree of freedom in honeycomb structure of two-dimensional crystals. Carriers in + k and − k (opposite Berry curvature) in transition metal dichalcogenide (TMD) with broken inversion symmetry act as effective magnetic fields, where this polarized valleys are suitable for encoding information. In this work, we study the strained TMD nanoribbons by Slater-Koster tight-binding model, which acquires electronic bands in whole Brillouin zone. From this, we derive a generic profile of strain effect on the electronic band structure of TMD nanoribbons, which shows indirect band gap, and also exhibits a phase transition from semiconductor to metallic by applying uniaxial X-tensile and Y-arc type of strain. Midgap states in strained TMD nanoribbons are determined by calculation of localized density of electron states. Moreover, our findings of anomalous valley Hall conductivity reveal that the creation of pseudogauge fields using strained TMD nanoribbons affect the Dirac electrons, which generate the new quantized Landau level. Furthermore, we demonstrate in strained TMD nanoribbons that strain field can effectively tune both the magnitude and sign of valley Hall conductivity. Our work elucidates the valley Hall transport in strained TMDs due to pseudo-electric and pseudo-magnetic filed will be applicable as information carries for future electronics and valleytronics.

To manipulate electron valley degree of freedom, a proper means can be VHE 28,[38][39][40] . Similar to an ordinary Hall effect, in which by applying a uniform magnetic field in real space cause to drive a transverse charge current, a transverse valley current in the VHE in k-space is created by valley-contrasting Berry curvatures 18,[41][42][43][44][45][46][47][48][49] , which traverse carriers in opposite direction from different valleys by the application of an external electric field. Therefore, 2D hexagonal materials with K and K′ valleys in the Brillouin zone exhibit VHE suitable for valleytronics 18 . Son et al. 50 applied strain to the monolayer TMD, which induces the Berry curvature dipole, enabling the mechanical tuning of valley magnetization due to in-plane electric field. In other study by Xu et al. 51 fully spinand valley-polarized anomalous Hall conductivity have been obtained in the WS 2 /MnO heterostructure as a result of large valley splitting and time-reversal symmetry broken.
In this work, we carry out systemic electronic structure calculations based on tight-binding (TB) approach to investigate the electronic properties and valley Hall transport in both X-tensile-and Y-arc strained TMD nanoribbons. We find that (1) as the uniaxial strain increases, TMD nanoribbons exhibit a transition from semiconductor to metal, where valance valleys are shifted towards positive energies. Also, Midgap states in strained TMD nanoribbons are tuned with the strain field. (2) The sign and magnitude of anomalous valley Hall conductivity (AVHC) of TMD nanoribbons has been changed with strain. (3) In TMD nanoribbons under strain, psedomagnetic field affects the Berry curvature and creates exotic surface states, while the evolution of Berry curvature correlates with nonmonotonic change of AVHC. These results altogether suggest that strain field play as a powerful technique for tuning the quantum electronic states as well as Berry curvature and AVHC applicable in a wide range of quantum advanced materials.

Methods and model description
In this study, we model six strained TMD nanoribbons i.e., MoS 2 , MoSe 2 , MoTe 2 , WSe 2 , WSe 2 and WTe 2 using TB model. A finite-size TMD nanoribbons in one direction is constructed, then we have to passive or remove a few dangling bonds on the edge of the system. These are not desired and we can remove these dangling bonds in our TB model by setting lattice neighbors attribute. In the TB model, we used the minimum lattice neighbors method, which is required to remove any atoms, which have less than the specified minimum number of neighbors. We consider two types of strained structure as represented in Fig. 1, as named as uniaxial X-tensile and uniaxial Y-arc strain with several displacement such as, δ = 0, 0.02, 0.04, 0.06, 0.08, 0.1, 0.15, 0.2. Six monolayer TMDs have a direct band gap at the K and K′ points of the hexagonal Brillouin zone (BZ), which behave as a semiconductor.
Ab-initio calculations reveal two additional secondary extrema, that a local minimum of conduction band (CB) is located at Q point, while a local maximum of the valence band (VB) is located at the Γ point, midway between Γ and K point 52 . These features are not consistent with their optical properties and transport 53,54 . The massive Dirac Hamiltonian describe the low-energy K and K′ points of monolayer MoS2 45 .
TB Hamiltonian 55-57 and k.p approximation 56,58 has been developed as accurate approximations beyond the massive Dirac model, which take into account for diagonal quadratic terms in momentum and the presence of trigonal warping. In this section, we employ TB Hamiltonian for calculating the electronic band structure of strained TMDs.
Tight binding model for strained TMDs. The Slater-Koster TB Hamiltonian in Ref. 59 captured the electronic band structure of monolayer MoS 2 in the whole BZ, including 11 bands of the d orbitals of the transition metal (Mo) and the p orbitals of the chalcogenide (S) atoms. It's worth to note that the physics of monolayer MX 2 around the band gap can be obtained by performing an unitary transformation in the subspace that transform the p orbitals of the bottom and top X layers into their combination of symmetric and antisymmetric with respect to the z-axis. For including the local spin-orbit interaction 60 , dominating of diagonal term L z S z can be appear, which each spin sector can be dealt with separately 60 . Figure 1 represents a top view of the crystal lattice of MX 2 . The reduced Hilbert space can be considered using compact notation of Ref 59 : where the S and A superscripts refer to symmetric and antisymmetric of the p-orbitals combination of where c † i,µ c i,µ creates (destroys) an electron in the atomic orbital of μ = 1,..,6 of Hilbert space of base (1) and in the unit cell i. A compact form of the TB Hamiltonian in the k-space is:  www.nature.com/scientificreports/ where the δ i and a i are the nearest and next nearest neighbor vectors are shown in Fig. 1. The hopping terms t ij,µν within a Slater-Koster approach have been considered [59][60][61] , which brought in Table 1.
Hamiltonian in strained lattice. The Slater-Koster TB approach for lattice deformations like strain is convenient 65 . In this approach, the effect of strain takes into account by considering of TB parameters of energy integral element of two-center energy dependent on the interatomic distances, which the correction to the local atomic potentials due to lattice deformation is neglected as a first approximation 63,64 . Here, we apply strain effect by varying the interatomic bond length in the simplest way. The modified hopping terms with strain at the linear order can be written as 65 : where |r 0 ij | is the distance in the absence of strain at the equilibrium positions between two atoms of (i,μ) and (j,ν), while |r ij | is the distance in the presence of strain. Here, � ij,µν = −dlnt ij,µν dln(r)|r =|r 0 ij | is the local electron-phonon coupling 65 . In practice, |r 0 ij | = a for in-plane M-M and X-X bonds and |r 0 ij | = 7 12 a for M-X bond have been applied 65 . In the absence of any theoretical and experimental estimation for the electron-phonon coupling, we use the Wills-Harrison argument 52 as t ij,µν (r) ∝ |r| −(l µ +l ν +1) , where l μ(ν) is the absolute value of the angular momentum of orbital μ(ν). Following this approach, ij,M−M = 5, ij,X−X = 3 for M-M dd and for X-X pp hybridization and ij,X−M = 4 for X-M pd hybridization. The vector r 0 as separation of two lattice site connected with electron hoping is transformed by application of strain into r ∝ r 0 + r 0 .∇u 65 , where ∇u = ε + ω ; ε is the strain tensor and ω is the rotation tensor. The strain tensor for 2D materials is a symmetric tensor as: with components including u ii is the in-plane and u ij is the out-of-plane displacement as: where u = (u x , u y , u z ) is the displacement vector and r = (x,y) is the position vector. To account the local rotation in the system, we use the ω as the anti-symmetric rotation tensor as defined: 2 ω xy = − 2 ω yx = ∂u y ∂x + ∂u x ∂y , which ω for homogenous strain will be zero. It is worth to note that the transformation relation is r = r 0 .(1 + ε) for homogenous strain and r = r 0 .(1 + ∇u) for inhomogeneous strain fields.
Hall conductivity. Valence band (VB) and conduction band (CB) edges of monolayer MoS 2 are located at the corners of K points of hexagonal plane 66 . The large separation of two inequivalent valleys in momentum space constitutes a binary index, which is robust against scattering by long wavelength phonons and deformation 45 . Therefore, coexistence of VHE in TMD monolayer has to be investigated similar to graphene. Broken inversion symmetry in TMD monolayers give rise to VHE with flowing carriers in different valleys by applying electric field. Calculation of quantum VHE of 2D electron gas indicates the quantized nature in unit of e 2 /ħ, which observed in graphene at room temperature 67 . In TMDs with time-reversal or broken inversion, pronounced Berry curvature can emerge VHE. Quantum Hall conductivity is arose due to anomalous velocity of electrons in the presence of an in-plane electric field, which is proportional to the Berry curvature in the transverse direction 7,68 , defined as: ; is the sum of band resolved of Berry curvature or alternative expression defines as � n

Results and discussions
Monolayer TMDs as a direct gap semiconductor emerge advanced optical materials for device applications. Hsu et al. 71 reported that strain modifies the wavefunctions, band curvature and optical matrix, which affects the binding energy of the K-K direct exciton and radiative lifetime. In this study, we determine the direct/indirect gap properties for TMD nanoribbons without/with strain fields, which play as an effective perturbation for modulating electronic properties. We have first studied the electronic structures of six compound of TMDs nanoribbons with applied X-tensile and Y-arc strain by the TB approach. Taking the MX 2 nanoribbons as an example, the model of strained structure is shown in Fig. 1.   78 . They showed that TMD nanoribbons with zigzag edges are always metallic regardless of the composition, the width or the edge structures, which is in good agreement with our results 78 . Furthermore, the effect of strain on the electronic and magnetic properties of the MoS 2 nanoribbons has been investigated by first-principle calculation 79 . Where, the stretchable MoS 2 is nonmagnetic and its band gap decreases with increasing strain, and direct band gap at weak strain changes to indirect band gap with increased strain up to 10% 79 , which are in good agreement with our results. The band gap response of MoTe 2 for both type of strain is the same, where it remains metallic. The CB minimum and the VB maximum of strained MoTe 2 are contributed and dominated from the d orbitals of Mo atoms and p orbitals of Te atoms 77,80 , which make the strain effect on band gap unambiguous. Figure 4 indicates the Electronic band structure of MoS 2 without strain and with two types of strain, where valance valleys are shifted towards positive energies by red arrows after external stimuli. Red dashed lines in this figure show the direct and indirect band gap.
In several reports have been revealed that the electronic and transport properties of TMD semiconductors can be crucially impacted by midgap states induced by dopants, defects, electric field and strain field, which can be native or intentionally incorporated in the crystal lattice [81][82][83] . The midgap states in the band structures of strained TMD nanoribbons in this study originate mainly from strain field, and tune with strain values. Herein, we calculated local density of states (LDOS) (commensurate with experimental STM data) to characterize the Herein, we apply the Slater-Koster TB-Hamiltonian to illustrate the strain effects on the electronic band structure (Eqs. 1, 2, 3) and the symmetry properties at specific k points relevant to the low-energy degrees of freedom, where modified hopping term is responsible for existence of midgap states. It's worth to note that an alternative way to include the coupling terms such as strain and electromagnetic fields to TB-Hamiltonian has been established based on the effective k.p Hamiltonian, which expands the full TB model as (H TB (k) = H 0 TB (k) + H strain ) 84 , where both methods take into account the midgap states as crucial impact on the electron transport properties of strained TMD nanoribbons.
Valley hall conductivity. Intriguing properties in quantum materials are contributed by Berry curvature; such as intrinsic anomalous Hall effect. Exploring the evolution of Berry curvature due to external stimulus could be lead to emergent quantum transport properties. As mentioned in method section, Berry curvature (Eqs. 8,9) is sensitive to changes of wavefunction and electronic band structure, which is tuned by external stimulus. For instance, in gated monolayer TMDs, controllable of VHE has been proposed by Rashba type spin orbit coupling, which needs strong displacement fields of 0.3-0.4 eV/A° in ionic liquid gated device [85][86][87] . We further determine that strain field can manipulate Berry curvature, where plays a significant role in exotic electronic www.nature.com/scientificreports/ states of quantum materials, such as the VHE. Herein, we apply two uniaxial strains to TMD nanoribbons for tuning of Berry curvature and investigate the changes of magnitude and sign of their valley Hall conductivity due to modified electronic structures. Our calculation of band structure of strained TMD nanoribbons reveals that valence bands are shifted to positive energies in the K-space. Therefore, strain alters the sign of valence bands and Berry curvature, which leads to sign and magnitude change of valley Hall conductivity as shown in Figs. The valley-carried orbital magnetic moment characterize the valley degree experimentally 7,88,89 . In our work, the orbital magnetic moment for conduction and valance band in the presence of the strain fields is defined as: where Ω is the Berry curvature and ν = ± stands for the valence and conduction band. This formula indicates that the magnitude and sign of magnetic moment depends on Berry curvature, which is tuned by strain field. Our findings reveal that strain field changes the order of d orbital energies of transition metal in TMD nanoribbons, which induces a crystal field splitting and lead to change of sign and magnitude of the valley Hall conductivity. We now discuss how strained TMDs affect the valley Hall conductivities due to Berry curvature features. We calculate the transverse (σ xx ) and valley Hall conductivity (σ xy ) of TMD nanoribbons without strain and with X-tensile strain as represented in Figs  www.nature.com/scientificreports/

Pseudoelectric and pseudomagnetic field (Landau level). Inhomogeneous strains in graphene can
induce pseudomagnetic fields very similarly to real fields 93 . The induced magnetic field introduce multiple singularities in density of states as Landau level, which is separated by band gaps 91 . A strain gradient creates pseudoelectric and pseudomagnetic fields at strained structure, where high or low density of atoms and, hence, electrons (inhomogeneous charge distribution) are emerged as shown in marked regions in Fig. 1, which results in an pseudoelectric field. In Y-arc strained TMD nanoribbons, stretching of bonds cause the momentum K and K′ points of Dirac cones shift as δk from their unstrained positions in the reciprocal space. This δk as a momentum shift generates a pseudovector potential term eA/c 92 , which creates opposite signs of pseudomagnetic fields at the two valleys. The Y-arc strain creates rare and dense regions in the TMD nanoribbons, acting as two different materials in a superlattice. Pseudomagnetic fields are ± ẑ field direction for pseudo spin up and down, where the valley polarized states are formed due to reversal of pseudospins.

Conclusion
In conclusion, by employing TB approach we have investigated the electronic, and valley Hall conductivity of six TMD nanoribbons such as MoS 2 , MoSe 2 , MoTe 2 , WSe 2 , WSe 2 and WTe 2 considering X-tensile and Y-arc strain up to 20%. The nature of electronic structure is indirect for both type of strained TMD nanoribbons and a transition from semiconductor to metallic is observed for almost all TMD nanoribbons by enhancing strain fields, where the valance valleys are shifted towards positive energies. Furthermore, we note that anomalous valley Hall conductivity (AVHC) of TMD nanoribbons in the sign and magnitude altered with strain. The deformed hexagonal structure of reciprocal lattice in Brillouin zone by strain field moves the Dirac cones at the K and K′ points (changed momentum K → K + δk) in opposite direction, which interpreted as a pseudo-vector potential. Meanwhile, pseudomagnetic field due to gradient strain in TMDs affects electron valleys in opposite direction for K and K′ valleys. Moreover, stretching the TMDs lattice as Y-arc strain changes the local electron density, which creates an in-plane electric field. Pseudomagnetic field in strained TMD nanoribbons affects the Berry curvature and emerges new quantized Landau level in align with nonmonotonic change of AVHC. Our findings demonstrate that there is a coupling between the strain field and the valley degree of freedom in TMD nanoribbons, which leads to a significant advance in valley-dependent electronics as well as fundamental condensed matter physics.  www.nature.com/scientificreports/ 91. Kane, C. L. & Mele, E. J. Size, shape, and low energy electronic structure of carbon nanotubes. Phys. Rev. Lett. 78, 1932Lett. 78, -1935Lett. 78, (1997